As a start let's show the correctness of some of the formulas on that page. Let's start with the following identity:

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It is correct. Now we want to make the left hand side look like the one provided in the article, so we want to make:

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Note that[LaTeX ERROR:
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which looks a lot like the complex definition for cosine, except that we need t to be [LaTeX ERROR:
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and instead of a minus in the middle, we need a plus. After a few tries, just plug in [LaTeX ERROR:
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, because we need a factor of [LaTeX ERROR:
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, in order to obtain the desired exponent.

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Here we've used that [LaTeX ERROR:
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and hence the plus sign between the exponential functions. Now, let's plug in [LaTeX ERROR:
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in the doubly infinite sum:

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Thus we have the first identity [LaTeX ERROR:
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For reasons I will explain later, we need to find an analogous identity for [LaTeX ERROR:
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(It has to do with the complex definitions of sine and cosine). Let's use the initial identity. This time, however, we want to make [LaTeX ERROR:
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, so just use [LaTeX ERROR:
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to get: